Question 3554: Least Counterexample in a Finite Domain

Original question and givens

Using classical truth semantics, the explicit finite integer domain is U={-1, 0, 1} and P(x) means x is an integer multiple of (2). Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

• G001 — Using classical truth semantics, the explicit finite integer domain is U={-1, 0, 1} and P(x) means x is an integer multiple of (2). Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-counterexample

Step-by-step derivation

U={-1, 0, 1};predicate=multiple 2

a counterexample requires c∈U and P(c)=False

test 2 divides -1

-1=-1·2+1

x=-1;predicate=multiple 2

P(-1)=No

test 2 divides 0

0=0·2+0

x=0;predicate=multiple 2

P(0)=Yes

test 2 divides 1

1=0·2+1

x=1;predicate=multiple 2

P(1)=No

[[-1, No], [0, Yes], [1, No]]

all finite-domain failures={-1, 1}

{-1, 1}

-1

the explicitly declared finite exercise

no inference about an unspecified infinite domain

Reverse checks

• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

-1

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.